Statistics

Introduction:

Statistics deals with data collected for specific purposes. We can make decisions about the data by analyzing and interpreting it. There are methods of finding a representative value for the given data. This value is called the measure of central tendency. The three measures of central tendency are mean, median and mode.

Mean: The mean or average of observations is the sum of the values of all the observations divided by the total number of observations.

Median: The median is the measure of central tendency which give us the value of the middle most observation in the data.

If n is odd, the median is the observation.

If n is even, the will be the average of the and the observation.

Mode: A mode is that value among the observation which occurs most often, that is, the value of the observation having the maximum frequency.

Measures of Dispersion:

The measure of dispersion shows the scatterings of the data. It tells the variation of the data from one another and gives a clear idea about the distribution of the data. The measure of dispersion shows the homogeneity or the heterogeneity of the distribution of the observations. The following are the measures of dispersion:

I. Range

II. Quartile deviation

III. Mean Deviation

IV. Standard deviation

Range:

It is the difference between two extreme observations of the data set. If X _max and X _minare the two extreme observations then

Range = X _max – X _min

Mean Deviation:

Mean deviation is the arithmetic mean of the absolute deviations of the observations from a measure of central tendency. If x₁, x₂, … , x_n are the set of observation, then the mean deviation of x about the average A (mean, median, or mode) is

Mean deviation from average A =

I. Mean Deviation for Ungrouped Data:

a. M.D.( where =Mean

Question 1: Find the mean deviation about the mean for the data given

38,70,48,40,42,55,63,46,54,44

Solution: Mean of the given data is

M.D.( where =Mean

Total

M.D. about mean=

M.D.( where M=Median

Question 2: Find the mean deviation about median for the data given

13,17,16,14,11,13,10,16,11,18,12,17

Solution: Arrange the data in ascending order, we have

10,11,11,12,13,13,14,16,16,17,17,18

Here, n=12(even number)

So median is average of 6^th and 7^th observations

Median=

M.D.( where M=Median

3.5

2.5

1.5

0.5

2.5

3.5

4.5

Total

M.D. about median=

II. Mean Deviation for Grouped Data:

a. Discrete Frequency Distribution: A discrete frequency distribution lists all the observed values.

i. Mean Deviation about Mean:

M.D.(

Question 3: Find the mean deviation about the mean for the data given

	5	10	15	20	25
	7	4	6	3	5

Solution: Mean (

125

350

158

M.D. about mean

ii. Mean Deviation about Median:

M.D.(

Question 4: Find the mean deviation about median for the data given

	5	7	9	10	12	15
	8	6	2	2	2	6

Solution:

The c.f. just greater than 13 is 14 and corresponding value of x is 7.

Therefore, Median=7

M.D. about median=

b. Continuous Frequency Distribution: It is a series in which the data are classified into different class interval without gaps along with their respective frequencies.

i. Mean Deviation about Mean: This is same as the discrete frequency distribution except that here represents the middle point of the class interval.

M.D.(

Question 5: Find the mean deviation about the mean for the data given

Income

per day

0-100

100-200

200-300

300-400

400-500

500-600

600-700

700-800

Number of persons

Solution:

Income per day

Mid-value

0-100

100-200

200-300

300-400

400-500

500-600

600-700

700-800

150

250

350

450

550

650

750

200

1200

2250

3500

3150

2750

2600

2250

308

208

108

192

292

392

1232

1664

972

644

960

1168

1178

17900

7896

Mean = =

M.D. about mean =

ii. Mean Deviation about Median:

Here, Median =

Where, l = lower limit,

N = sum of frequencies,

f= the frequency,

C = cumulative frequency,

h = width of the median class.

Then, M.D. (

Question 6: Find the mean deviation about median for the data given

Marks	0-10	10-20	20-30	30-40	40-50	50-60
Number of Girls	6	8	14	16	4	2

Solution:

Marks

Mid value

c.f.

0-10

10-20

20-30

30-40

40-50

50-60

22.86

12.86

2.86

7.14

17.14

27.14

137.16

102.88

40.04

114.24

68.56

54.28

517.16

Median class is 20-30

Median= 20+

M.D. about median

III. Limitations of Mean Deviation:

i. Not easily understandable.

ii. Its calculation is not easy and time-consuming.

iii. Dependent on the change of scale.

iv. Ignorance of negative sign creates artificiality and becomes useless for further mathematical treatment.

Variance:

Mean of the square of the squares of the deviations from mean is called the variance.

Variance,

Standard Deviation:

A standard deviation is the positive square root of the arithmetic mean of the squares of the deviations of the given values from their arithmetic mean. It is denoted by a Greek letter sigma, σ. It is also referred to as root mean square deviation. The standard deviation is given as

Standard Deviation,

Question 7: Find the variance and standard deviation deviation of the following data

3, 4, 6, 5, 5, 3, 8, 1, 7, 5

Solution: Here x=3, 4, 6, 5, 5, 3, 8, 1, 7, 5.

n=10

Therefore,

=259

S.D. = = = =

Variance =

I. Standard Deviation of a Discrete Frequency Distribution:

Let the given discrete frequency distribution be

Standard Deviation (

where, N=

Question 7: Find the variance and standard deviation deviation of the following data

	2	4	6	8	10	12	14	16
	4	4	5	15	8	5	4	5

Solution:

120

-7

-5

-3

-1

196

100

245

450

754

Mean,

Variance, =

Standard Deviation,

II. Standard Deviation of a Continuous Frequency Distribution:

If there is a frequency distribution of n class defined by its mid-point with frequency , the standard deviation will be obtained by the formula

where,

III. Another Formula for Standard Deviation:

Standard Deviation (

Question 8: Find the mean,variance and standard deviation of the following frequency distribution

Class	0-30	30-60	60-90	90-120	120-150	150-180	180-210
Frequencies	2	3	5	10	3	5	2

Solution:

Classes

Mid values

0-30

30-60

60-90

90-120

120-150

150-180

180-210

105

135

165

195

-3

-2

-1

-6

-5

Mean, A+

Variance, =

Standard deviation,

Analysis of Frequency Distributions:

The measure of variability which is independent of units is called coefficient of variation.

Coefficient of Variation is defined as

C.V.=

where and are the standard deviation and mean of data.

I. Comparison of Two Frequency Distribution with Same Mean:

Let and be the mean and standard deviation of the first distribution, and and be the mean and standard deviation of the second distribution.

C.V.(1^st distribution)=

C.V.(2^nd distribution)=

Given ==

Then,

C.V.(1^st distribution)=

C.V.(2^nd distribution)=

From above it is clear that the two C.V.s can be compared on the basis of values of and only,

Thus, we say that for two series with equal means, the series with greater S.D. (or variance) is called more variable or dispersed that the other. Also, the series with lesser value of S.D. (or variance) is said to be more consistent than the other.

Question 9: From the prices of shares X and Y below, find out which is more stable in value: